- Research
- Open Access
Group-based single image super-resolution with online dictionary learning
- Xuan Lu^{1},
- Dingwen Wang^{2}Email authorView ORCID ID profile,
- Wenxuan Shi^{3} and
- Dexiang Deng^{1}
https://doi.org/10.1186/s13634-016-0380-9
© The Author(s) 2016
Received: 20 February 2016
Accepted: 14 July 2016
Published: 29 July 2016
Abstract
Recently, sparse representation has been successfully used in single image super-resolution reconstruction. Unlike the traditional single image super-resolution methods such as image interpolation, the super-resolution with sparse representation reconstructs image with one or several constant dictionaries learned from external databases. However, the contents can vary significantly across different patches in a single image, and the fixed dictionaries cannot suit for every patch. This paper presents a novel approach for single image super-resolution based on sparse representation, which uses group as the basic unit, and trains dictionary with external database and the input low-resolution image itself for each group to ensure that the dictionary is suitable for the patches in the group. Simultaneous sparse coding algorithm is used to accelerate the processing and improve the result. Extensive experiments on natural images show that our method achieves better results than some state-of-the-art algorithms in terms of both objective and human visual evaluations.
Keywords
1 Introduction
where H represents the blurring process and S represents the down-sampling process. ν is the additive noise. Super-resolution solves the inverse problem of the degradation while it remains extremely ill-posed, which means there are generally multiple solutions that can be degenerated to the same LR image.
where J(X) is a regularization term specifying the prior knowledge of the HR image and λ is a scalar balancing between the quadratic fidelity term and the regularization term, such as the total variation (TV) regularization [6], edge smoothness [7], and gradient profile priors [8]. However, these methods cannot recover fine details and have unnatural edges.
In the past several years, sparsity has been emerging as one of the most significant properties of natural images [9]. The sparsity prior suggests that image patch can be well-represented as a sparse linear combination of elements from an appropriately chosen over-complete dictionary [10]. The sparsity-based regularization has achieved great success both qualitatively and quantitatively. However, it still has a little jaggy and ringing artifact along the edges in the reconstructed image. One of the keys to improve the result is to find a more suitable dictionary. Different improvements were proposed [11–14], etc., and have gotten better results.
Another significant property exhibited in natural images is nonlocal self-similarity, which is based on an observation that patches in a single natural image tend to redundantly recur many times inside the image, both within the same scale, as well as across different scales [15]. In recent works, the sparsity and the self-similarity of natural images are usually combined to achieve better performance [13, 16, 17].
Traditional algorithms which are mentioned above often use patch as the basic unit of sparse representation and train redundant dictionaries with fixed sample image sets. Zhang et al. [18] and Zhang et al. [19] exploit the concept of group-based sparse representation for general image inverse problem and develop an efficient and effective algorithm for image restoration and image compressive sensing recovery. Inspired by these works, this paper uses group as the basic unit for image super-resolution. The main contribution of our proposed method is that we divide the input image into several groups to combine the sparsity and the self-similarity of natural images in a unified framework and improve the performance of the dictionary for each group with the novel online dictionary learning method, which is more suitable than the one trained with classic algorithms. Experiments show that the proposed algorithm outperforms many current state-of-the-art schemes.
The rest of the paper is organized as follows. Section 2 introduces the related works. Section 3 presents the proposed super-resolution method and gives its implementation details. Section 4 shows various comparison experiments. Section 5 gives the conclusions and discussions.
2 Background and preliminaries
2.1 Super-resolution via sparse representation
where parameter γ balances the fidelity term and the sparsity of the solution and F is a feature extraction operator.
which is known in statistical literature as the lasso.
where 1 _{ n } is a vector of size n with all its elements being 1 and N is the total amount of the patches.
where A denotes the concatenation of all α _{ k }, i.e., A=[α _{1},α _{2},…,α _{ N }].
where γ is the regularization parameter. With \(\hat {\mathbf {A}}\), the reconstructed image can be expressed by \(\hat {\mathbf {X}} = \mathbf {D}_{h}\circ \hat {\mathbf {A}}\).
where λ balances between the fidelity term and the regularization term.
2.2 Dictionary learning
where Z is the set of sparse representations of the training set X and the l _{1}-norm ||Z||_{1} is used for enforcing sparsity. Eq. (10) is not convex in both D and Z but is convex in one of them when the other is fixed. So, it can be solved in an alternative manner over Z and D.
Thus, the strategy of single dictionary learning can be used for training the two dictionaries for SR purpose.
3 The proposed algorithm
The non-local similarity prior for natural images is based on an observation that patches in a single natural image tend to redundantly recur many times inside the image. On the other hand, natural images are believed to be composed of simple local image structures, observed as singular primitives, such as lines and arcs [3]. These local singular primitives are invariant to scale changes. So, a patch has good matches around its original location in the lower scale image [22].
These researches show that an natural image can be divided into several groups. The patches in the same group have similar image structures and can be presented by a relatively compact dictionary, which is more suitable for the patches in the group than a redundant dictionary. We use group as the basic unit instead of patch to gain a better result at the same time.
We treat the input LR image as the image that contains some high-frequency contents but with unsatisfactory pixel resolution. So, it offers high-frequency information about the singular primitives and can be used in our group-based SR algorithm. Let \(\mathbf {X}_{l}\in \mathbb {R}^{K}\) denote the input LR image, where K is the size of the whole image vector. We down-sample X _{ l } and then up-sample it using bi-cubic interpolation by the same factor of s to obtain the low-frequency band image \(\mathbf {Y}_{l} \in \mathbb {R}^{K}\). Then, we up-sample X _{ l } with bi-cubic interpolation by the factor of s to obtain low-frequency band \(\mathbf {Y}_{h} \in \mathbb {R}^{s^{2}K}\) of the unknown HR image \(\mathbf {X}_{h} \in \mathbb {R}^{s^{2}K}\). Use \(\mathbf {y}_{l}^{k}\), \(\mathbf {y}_{h}^{k}\), \(\mathbf {x}_{l}^{k}\), and \(\mathbf {x}_{h}^{k}\) to denote the vector representations of the image patch extracted from Y _{ l }, Y _{ h }, X _{ l }, and X _{ h } in the kth position, respectively.
Incorporating with the nonlocal similarity prior knowledge, for a patch \(\mathbf {y}_{h}^{k}\), we search the similar patches around the corresponding place and make a group, which is denoted by \(G_{y_{h}}^{j}\), where j is the group order, and the total number of the group is denoted by L. Euclidean distance is selected as the similarity criterion between different patches. The number of the patches in the group is c, that is to say we choose c−1 patches that are most like the patch \(\mathbf {y}_{h}^{k}\) to make a group, and then delete them from the patch list. So \(L = \left \lceil \frac {P}{c}\right \rceil \), where P is the total number of the patches and ⌈·⌉ is the ceiling function. The corresponding patches of \(\mathbf {y}_{h}^{i} \in G_{y_{h}}^{j} (i = 1, 2, \ldots, c)\) in image Y _{ l } can also make a group, which is denoted by \(G_{y_{l}}^{j}\). The group \(G_{y_{l}}^{j}\) and its high-frequency version \(G_{x_{l}}^{j}\) provide the information about the lost high-frequency band to the unknown image X _{ h }.
3.1 Online dictionary learning phase
In this paper, online dictionary learning is imported to train suitable dictionary for the group. Online dictionary learning was first presented by [23], which can handle potentially infinite data sets, adapt to dynamic training sets, and it is dramatically faster than traditional algorithms. Instead of using a fixed dictionary, we try to update dictionary during the image processing of each patch. The patches in group \(G_{x_{l}}^{j}\) can be treated as the corresponding patches in group \(G_{y_{l}}^{j}\) with high frequency. And the group \(G_{y_{l}}^{j}\) and the group \(G_{x_{l}}^{j}\) compose the training set.
To ensure that the HR dictionary D _{ h } and the LR dictionary D _{ l } have the same sparse representation, we use the method mentioned in Section 2.2 and transform the joint dictionary learning into a single one.
where the superscript “T” means transpose. In this paper, we use Eq. (14) as F due to its simplicity and effectiveness.
where N and M are the dimensions of the HR and LR image patches in vector form.
where \(\mathbf {x}_{i}^{j}\) is the sample vector with index i in the group set \({G_{c}^{j}}\) and D _{0} is an initial dictionary which is learned using an external database. Thus, we have finished learning dictionary D ^{ j } for the patches in the group with the index j.
The basic unit of dictionary learning is still patch, but each dictionary is only updated by the patches in the corresponding groups. Different from the traditional dictionary learning method proposed by Yang et al. [10] that uses a single dictionary for the construction of all patches, this method also contained the information of all patches in the group in dictionary learning phase, which made the dictionary more suitable for the patches in the corresponding group. Because the patches in the group are similar, the dictionary is relatively more compact than the initial one.
We randomly extracted 80,000 patches from 200 high-quality natural images from the Berkeley Segmentation Database [26]. With the same feature extractor F mentioned above and the method introduced in Section 2.2, we can calculate D _{0}.
3.2 Simultaneous sparse coding phase
The sparse coding phase attempts to magnify all of the patches in the input LR image X _{ l }. After estimating the low-frequency part Y _{ h } of the final HR image X _{ h }, we just need to restore the high-frequency part and then add it to Y _{ h }.
We apply simultaneous sparse coding to the sparse representation of each group. For traditional sparse coding, similar patches in one group sometimes admit very different estimates due to the potential instability of sparse decomposition, which can result in noticeable reconstruction artifacts [27]. A simultaneous sparse coding algorithm makes approximation of several input signals at the same time using different linear combinations of the same elementary signals [28]. It solves the problem of the traditional sparse coding by forcing similar patches to admit similar decomposition.
where α _{ i } is the ith row of A. In practice, the value of the pair (p,q) is usually chosen as (1,2) or (0,∞), the former leading to a convex norm, while the latter actually counts the number of nonzero rows.
This optimization can be solved by simultaneous orthogonal matching pursuit (S-OMP) algorithm [28].
Our method skips the back-project step mentioned in (9), because the sparsity prior is strong enough that we can already achieve good performance.
Algorithm 1 shows the complete process of our proposed method.
4 Experimental results
We compare the proposed method with other four algorithms to illustrate the efficiency of our proposed method. The competed algorithms are bi-cubic interpolation [13, 15, 20]. Specifically, for methods based on fixed external dictionaries, we choose the work of Yang et al. [20] for comparison; for methods based on non-local similarity, we choose the work of Glasner et al. [15] for comparison; for methods that combine sparse representation and non-local similarity, we choose the representative work ASDS method [13] for comparison.
A frequently used criterion, peak signal-to-noise ratio (PSNR), is used for the image quality analysis. But it is sometimes not a reliable metric for evaluating the image quality. Therefore, the structural similarity (SSIM) index [29] and the feature similarity (FSIM) index [30] are also adopted for the objective evaluation. A higher PSNR value implies less distortion compared with the ground truth, and an SSIM value or an FSIM value much closer to 1 indicates the structure or the feature of the reconstructed image is more similar to the ground truth image, respectively.
For color image super-resolution, we only apply our algorithm on the illuminance component and use bi-cubic interpolation for the chromatic components, because human visual system is not sensitive to the chromatic components. In experiments, the value of PSNR, SSIM, and FSIM are all conducted on the illuminance component of the image.
4.1 Experimental configuration
We magnify the input LR image by factor of 2 and use 6×6 patches with an overlap of two pixels between adjacent patches, both for the HR image Y _{ h } and X _{ h } and LR image Y _{ l } and X _{ l }; and we learned the dictionary of size K=128. In the online dictionary learning phase, the size L of training window is selected as 60, the number c of best matched patches is 128, the sparsity regularization parameter λ is 0.15, and the number T of iteration to train the dictionary is 32. In the simultaneous sparse coding phase, the parameter γ that balances the fidelity term and the regularization term is 0.15, and the value of (p,q) is chosen as 1,2.
We use the images from the Berkeley Segmentation Dataset and Benchmark [26] to train dictionaries for our method and the method of Yang et al. [20]. Because there are large differences between our method and the methods being compared, we just use the default parameters as configurations of these methods.
The proposed algorithm is implemented by MATLAB R2011b using SPAMS toolbox [31] for on-line dictionary learning and simultaneous sparse coding. The computer system used for simulation is Intel Core i7-4500U CPU at 1.80GHz with 8GB of RAM.
4.2 Noiseless experiment
4.2.1 Objective evaluations
Comparisons of peak signal-to-noise ratio(PSNR) values, structural similarity (SSIM) values, and feature similarity (FSIM) values for 15 test images with different super-resolution approaches
Image | Bi-cubic | Yang et al. [20] | Glasner et al. [15] | ASDS [13] | Proposed | |
---|---|---|---|---|---|---|
Avion | PSNR | 27.01 | 29.13 | 28.48 | 3 1.6 2 | 30.92 |
SSIM | 0.875 | 0.905 | 0.918 | 0.934 | 0.9 3 6 | |
FSIM | 0.863 | 0.902 | 0.907 | 0.930 | 0.9 3 6 | |
Barnfall | PSNR | 29.16 | 30.14 | 29.88 | 3 0.7 6 | 30.51 |
SSIM | 0.716 | 0.780 | 0.763 | 0.783 | 0.7 8 8 | |
FSIM | 0.825 | 0.895 | 0.862 | 0.893 | 0.8 9 9 | |
Blueeye | PSNR | 33.39 | 34.00 | 35.28 | 36.37 | 3 6.7 5 |
SSIM | 0.923 | 0.923 | 0.939 | 0.939 | 0.9 4 4 | |
FSIM | 0.946 | 0.957 | 0.960 | 0.962 | 0.9 6 8 | |
Butterfly | PSNR | 28.51 | 30.13 | 29.85 | 3 2.2 2 | 31.51 |
SSIM | 0.855 | 0.894 | 0.902 | 0.921 | 0.9 2 6 | |
FSIM | 0.909 | 0.930 | 0.935 | 0.947 | 0.9 5 0 | |
Cactusflower | PSNR | 25.48 | 26.67 | 26.34 | 2 7.4 7 | 26.95 |
SSIM | 0.658 | 0.763 | 0.728 | 0.749 | 0.7 6 4 | |
FSIM | 0.793 | 0.877 | 0.844 | 0.863 | 0.8 8 5 | |
Cameraman | PSNR | 25.55 | 26.83 | 26.68 | 2 8.6 8 | 27.94 |
SSIM | 0.827 | 0.862 | 0.869 | 0.892 | 0.8 9 9 | |
FSIM | 0.821 | 0.883 | 0.864 | 0.9 0 8 | 0.901 | |
Colomtn | PSNR | 27.44 | 28.33 | 27.93 | 2 8.8 0 | 28.67 |
SSIM | 0.685 | 0.747 | 0.729 | 0.749 | 0.7 5 7 | |
FSIM | 0.807 | 0.877 | 0.841 | 0.876 | 0.8 8 6 | |
Desert | PSNR | 24.10 | 24.86 | 24.89 | 2 5.5 9 | 25.41 |
SSIM | 0.677 | 0.757 | 0.737 | 0.780 | 0.7 8 6 | |
FSIM | 0.812 | 0.879 | 0.852 | 0.890 | 0.8 9 6 | |
Frog | PSNR | 31.42 | 32.74 | 32.78 | 3 4.4 5 | 34.27 |
SSIM | 0.895 | 0.920 | 0.927 | 0.937 | 0.9 4 4 | |
FSIM | 0.894 | 0.931 | 0.924 | 0.945 | 0.9 5 3 | |
Goldgate | PSNR | 33.05 | 34.46 | 34.05 | 3 5.4 8 | 35.10 |
SSIM | 0.873 | 0.899 | 0.907 | 0.908 | 0.9 1 2 | |
FSIM | 0.877 | 0.919 | 0.909 | 0.922 | 0.9 2 5 | |
House | PSNR | 26.53 | 28.40 | 28.14 | 3 0.3 3 | 29.61 |
SSIM | 0.817 | 0.863 | 0.865 | 0.892 | 0.8 9 4 | |
FSIM | 0.835 | 0.886 | 0.888 | 0.913 | 0.9 1 9 | |
London | PSNR | 29.27 | 31.32 | 30.34 | 3 3.2 8 | 33.13 |
SSIM | 0.850 | 0.894 | 0.894 | 0.915 | 0.9 1 9 | |
FSIM | 0.837 | 0.903 | 0.877 | 0.9 1 9 | 0.918 | |
Lostlake | PSNR | 27.60 | 28.81 | 28.37 | 2 9.7 1 | 29.30 |
SSIM | 0.749 | 0.806 | 0.795 | 0.819 | 0.8 2 4 | |
FSIM | 0.839 | 0.896 | 0.877 | 0.903 | 0.9 0 7 | |
Parrot | PSNR | 28.93 | 31.21 | 30.19 | 3 4.0 5 | 32.92 |
SSIM | 0.911 | 0.926 | 0.939 | 0.945 | 0.9 4 9 | |
FSIM | 0.917 | 0.948 | 0.938 | 0.9 5 9 | 0.957 | |
Redrock | PSNR | 27.33 | 28.51 | 28.12 | 2 9.3 5 | 29.11 |
SSIM | 0.773 | 0.834 | 0.820 | 0.847 | 0.8 5 1 | |
FSIM | 0.820 | 0.889 | 0.858 | 0.8 9 4 | 0.893 |
The SSIM and FSIM value, which are more accurate than PSNR value to evaluate the image quality, simulate the characteristic of human visual system. The improvement of our results in SSIM and FSIM shows the structure and feature restoration are better than the other competing methods.
4.2.2 Visual quality evaluations
Our algorithm uses the dictionary trained from external databases to offer the missing high frequency and the input image itself to offer the ground truth information. The online learning updates the dictionary for each patch to combine the information of external databases and the ground truth. The updated dictionary is more suitable for the patch and avoids adding some artifacts to the restored image.
It is obvious that the results of the proposed method have better details and edges. For long edges, there exist many similar patches along the edge which make the updated dictionary more precise, so the edges of the proposed results are much sharper than those of other algorithms. For the details, the input image offers important information about the details, especially the repeat similar details, to our super-resolution, while the fixed dictionaries cannot. However, the very short edges and the non-redundant details can hardly find enough similar patches. With the fixed size of the search window, many patches that are not so similar are also added into the training sample set. This may mislead the dictionary updating, thus leads to a result that is not so fine. In addition, when the low-frequency part of the input image is too blurry, it also misleads the dictionary updating and produces blurry details.
In Fig. 4, the edges and the streaks of the petal of our method are clearer than that of Glasner et al. [15] and ASDS [13], while there is no additional noise and ringing artifact as which in the result of Yang et al. [20]. In Fig. 5, the fine lines in the wings of the butterfly are restored very well, and the round dot is not distorted when compared with the first three methods. Because the textures in the butterfly wings are repeated, which is convenient for us to collect much useful ground truth information about these details. The same reason is for the restoration of Fig. 6. The short stripes on the parrot’s face are separated in the results of bi-cubic, Yang et al. [20], ASDS [13], and our method, while the edges of other stripes in our result are sharper than that of bi-cubic and ASDS [13] and are more delicate than that of Yang et al. [20]. In Fig. 7, the details of the camera is restored better than the others, especially the white fine line in the left bottom corner of the camera. Besides, it is obvious that the edges of the cameraman’s coat are sharper than others. These indicate that our algorithm can recover fine details and sharp edges at the same time.
4.3 Noisy experiment
Comparisons of peak signal-to-noise ratio(PSNR) values, structural similarity (SSIM) values, and feature similarity (FSIM) values for image with different noise level with different super-resolution approaches
Noise level | Bi-cubic | Yang et al. [20] | Glasner et al. [15] | ASDS [13] | Proposed | |
---|---|---|---|---|---|---|
σ _{ ν }=0 | PSNR | 28.93 | 31.21 | 30.19 | 3 4.0 5 | 32.92 |
SSIM | 0.911 | 0.926 | 0.939 | 0.945 | 0.9 4 9 | |
FSIM | 0.917 | 0.948 | 0.938 | 0.9 5 9 | 0.957 | |
σ _{ ν }=1 | PSNR | 28.92 | 31.13 | 30.17 | 31.34 | 3 2.9 2 |
SSIM | 0.910 | 0.918 | 0.935 | 0.908 | 0.9 4 7 | |
FSIM | 0.917 | 0.945 | 0.937 | 0.934 | 0.9 5 6 | |
σ _{ ν }=3 | PSNR | 28.83 | 30.49 | 29.94 | 31.24 | 3 2.6 7 |
SSIM | 0.897 | 0.863 | 0.908 | 0.907 | 0.9 3 4 | |
FSIM | 0.913 | 0.918 | 0.925 | 0.936 | 0.9 5 2 | |
σ _{ ν }=5 | PSNR | 28.66 | 29.41 | 29.46 | 30.96 | 3 2.1 8 |
SSIM | 0.875 | 0.779 | 0.860 | 0.896 | 0.9 1 1 | |
FSIM | 0.904 | 0.872 | 0.901 | 0.934 | 0.9 4 1 |
From the table and the figure, we can see that the results of Yang et al. [20] and Glasner et al. [15] enhanced the noise. The result of ASDS [13] performs the best on noise suppressing, but this noise suppression also affects the image reconstruction and makes the result image more blurry than its result of noiseless image. The proposed method achieves a good result both on noise suppressing and image reconstruction. The proposed method is better than the other because the group-based dictionary learning phase gets a relatively compact and suitable dictionary for the patches in the group to be reconstructed. The sparse coding in each iteration in the online dictionary learning method helps suppress the noise in the training patches. Thus, the elements in the dictionary and the structures of the group have a correlation between them, while the elements are independent with the noise. Combined with the simultaneous sparse coding phase, the noise is suppressed and the structure and the details are retained.
4.4 Time comparisons
In this subsection, we compare the running time of the four methods. We have ignored the time of training dictionaries for Yang et al. [20] and ASDS [13] and the time of training initial dictionaries for the proposed method.
5 Conclusions
This paper describes a novel method of group-based single image super-resolution. With the property of non-local self-similarity, we divide the input image into several groups. Combining with the information from external databases, we train suitable dictionaries for each group using online dictionary learning method. Simultaneous sparse coding algorithm is used to accelerate the processing and improve the result. Experiments show that the proposed method can restore sharp edges and fine details and achieve good result on noise suppressing. The running time is comparable with other state-of-the-art algorithms.
In this paper, we just use the traditional Euclidean distance for the searching of the similar patches to make a group. For further research, we will focus on developing a evaluation that directly measure the probability between two patches that belong to the same group to improve the performance of the proposed method.
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 61501334).
Availability of data and materials
The images supporting the conclusions of this article are available in the “Test Images of Computer Vision Group”, All of the images are Copyright free. http://decsai.ugr.es/cvg/index2.php.
Authors’ contributions
XL and DW designed and carried out the experiments. XL and DD analyzed the experimental results. XL wrote the manuscript. WS and DD gave the critical revision and final approval. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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